cos2 (problem 3.4.1)

Percentage Accurate: 50.8% → 99.8%

Time: 9.0s

Alternatives: 5

Speedup: 107.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 50.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ (sin x) x) (/ (tan (/ x 2.0)) x)))

C

double code(double x) {
	return (sin(x) / x) * (tan((x / 2.0)) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) / x) * (tan((x / 2.0d0)) / x)
end function

Java

public static double code(double x) {
	return (Math.sin(x) / x) * (Math.tan((x / 2.0)) / x);
}

Python

def code(x):
	return (math.sin(x) / x) * (math.tan((x / 2.0)) / x)

Julia

function code(x)
	return Float64(Float64(sin(x) / x) * Float64(tan(Float64(x / 2.0)) / x))
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) / x) * (tan((x / 2.0)) / x);
end

Wolfram

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 50.0%

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

   2. div-inv 50.0%

      \[\leadsto \frac{\color{blue}{\left(1 \cdot 1 - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]

   3. metadata-eval 50.0%

      \[\leadsto \frac{\left(\color{blue}{1} - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

   4. pow2 50.0%

      \[\leadsto \frac{\left(1 - \color{blue}{{\cos x}^{2}}\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

4. Applied egg-rr 50.0%

   \[\leadsto \frac{\color{blue}{\left(1 - {\cos x}^{2}\right) \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]
5. Step-by-step derivation

   1. associate-*r/ 50.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 - {\cos x}^{2}\right) \cdot 1}{1 + \cos x}}}{x \cdot x} \]

   2. *-rgt-identity 50.0%

      \[\leadsto \frac{\frac{\color{blue}{1 - {\cos x}^{2}}}{1 + \cos x}}{x \cdot x} \]

6. Simplified 50.0%

   \[\leadsto \frac{\color{blue}{\frac{1 - {\cos x}^{2}}{1 + \cos x}}}{x \cdot x} \]
7. Step-by-step derivation

   1. unpow2 50.0%

      \[\leadsto \frac{\frac{1 - \color{blue}{\cos x \cdot \cos x}}{1 + \cos x}}{x \cdot x} \]

   2. 1-sub-cos 76.4%

      \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x \cdot x} \]

8. Applied egg-rr 76.4%

   \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x \cdot x} \]
9. Step-by-step derivation

   1. associate-/l* 76.5%

      \[\leadsto \frac{\color{blue}{\sin x \cdot \frac{\sin x}{1 + \cos x}}}{x \cdot x} \]

   2. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \frac{\frac{\sin x}{1 + \cos x}}{x}} \]

   3. hang-0p-tan 99.8%

      \[\leadsto \frac{\sin x}{x} \cdot \frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{x} \]

10. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}} \]
11. Add Preprocessing

Alternative 2: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \cos x}{x}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (/ (+ -1.0 (cos x)) x) (- x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((-1.0 + cos(x)) / x) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0052d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (((-1.0d0) + cos(x)) / x) / -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((-1.0 + Math.cos(x)) / x) / -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0052:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = ((-1.0 + math.cos(x)) / x) / -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(-1.0 + cos(x)) / x) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0052)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = ((-1.0 + cos(x)) / x) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 + N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 35.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot -0.041666666666666664} \]

   if 0.0051999999999999998 < x

   1. Initial program 97.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 99.4%

         \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

      2. div-inv 99.3%

         \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]

   4. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. *-commutative 99.3%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}} \]

      2. frac-2neg 99.3%

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{-\left(1 - \cos x\right)}{-x}} \]

      3. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{x} \cdot \left(-\left(1 - \cos x\right)\right)}{-x}} \]

   6. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\frac{-1 + \cos x}{x}}{-x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0052d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0052:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0052)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 35.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot -0.041666666666666664} \]

   if 0.0051999999999999998 < x

   1. Initial program 97.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.5e+76) 0.5 (* (/ 1.0 x) (- (/ 1.0 x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 7.5e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * ((1.0 / x) - (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 7.5d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x) * ((1.0d0 / x) - (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 7.5e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * ((1.0 / x) - (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.5e+76:
		tmp = 0.5
	else:
		tmp = (1.0 / x) * ((1.0 / x) - (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.5e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(1.0 / x) - Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.5e+76)
		tmp = 0.5;
	else
		tmp = (1.0 / x) * ((1.0 / x) - (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.5e+76], 0.5, N[(N[(1.0 / x), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999995e76

   1. Initial program 40.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{0.5} \]

   if 7.4999999999999995e76 < x

   1. Initial program 97.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

      2. div-inv 99.7%

         \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. div-sub 99.7%

         \[\leadsto \color{blue}{\left(\frac{1}{x} - \frac{\cos x}{x}\right)} \cdot \frac{1}{x} \]

      2. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(-\frac{\cos x}{x}\right)\right)} \cdot \frac{1}{x} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(-\frac{\cos x}{x}\right)\right)} \cdot \frac{1}{x} \]
   7. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\frac{1}{x} - \frac{\cos x}{x}\right)} \cdot \frac{1}{x} \]

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\left(\frac{1}{x} - \frac{\cos x}{x}\right)} \cdot \frac{1}{x} \]

   9. Taylor expanded in x around 0 66.2%

      \[\leadsto \left(\frac{1}{x} - \color{blue}{\frac{1}{x}}\right) \cdot \frac{1}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.5% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 50.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 52.4%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))